1. Development of an Atmospheric Climate Model with Self-Adapting Grid and Physics Joyce E. Penner 1, Michael Herzog 2, Christiane Jablonowski 3, Bram van Leer 1, Robert C. Oehmke 1, Quentin F. Stout 1, and Kenneth G. Powell 1 1 University of Michigan, Ann Arbor, MI; 2 Geophysical Fluid Dynamics Laboratory, 3 National Center for Atmospheric Research Introduction Adaptive grid library The goal of this research project is to develop adaptive grid techniques for future climate model and weather predictions. This approach will lead to new insights into small-scale and large-scale flow interactions that are unresolved by current uniform-grid simulations. Adaptive mesh refinement (AMR) techniques provide an attractive framework for atmospheric motions since they allow improved horizontal resolution in a limited region without requiring a fine grid resolution throughout the entire model domain. Therefore, the model domain to be resolved with higher resolution is kept at a minimum, greatly reducing computer memory and speed requirements. Adaptive grid techniques have been developed for a parallel version of the NASA / NCAR Finite-Volume Community Climate Model. This global hydrostatic model is based on NCAR physics and the so-called Lin-Rood finite-volume dynamical core that provides highly efficient algorithms for high performance computing. This research project is characterized by an interdisciplinary approach involving atmospheric science, computer science and mathematical/numerical aspects. The work is done in close collaboration between the Atmospheric Science, Computer Science and Aerospace Engineering Departments at the University of Michigan and NASA. The newly developed version of the NASA finite-volume dynamical core with self-adaptive grid uses a parallel program library for block-wise adaptive grids on the sphere. As indicated in figure 1, the grid is subdivided horizontally into self- similar blocks that contain an identical number of grid points per block. In the event of a refinement request a block is split into four new blocks thereby doubling the spatial resolution. Here the spatial resolution of adjacent blocks is only allowed to differ by a factor of two. On the sphere, a regular longitude- latitude grid has been adopted. In addition, an initial reduced grid setup can be selected (Figure 2). In case the reduced grid is selected, the longitudinal resolution in polar regions is coarsened which alleviates the convergence of the meridians at the poles. The Hydrostatic Dynamical Core Statically and dynamically adaptive grids have been successfully implemented and tested in 2D shallow water simulations and 3D hydrostatic dynamical core runs on the sphere. Figure 3 shows an example of a 2D shallow water simulation at model day 10. The depicted geopotential height field is characterized by a lee-side wave that is induced by an idealized mountain. Here a combination of statically and dynamically refined blocks is presented. The dynamic adaptations track the evolution of the wave by a gradient-based adaptation criterion. Other adaptation criteria that are, for example, based on vorticity have also been successfully applied. This project is aimed at developing a climate model that self-adjusts the grid resolution and the complexity of the physics model to the actual atmospheric flow conditions. To accomplish this, we have implemented a fully 3-dimensional non- hydrostatic model within a hydrostatic code while using a block-structured grid that allows for the implementation of smaller grid resolution within both the hydrostatic and non-hydrostatic portions of the grid. where is the geopotential, is the pressure and is the horizontal velocity. The full density is used to predict the full geopotential, from which the non- hydrostatic geopotential anomaly is derived, and the hydrostatic vertical velocity is derived from the temporal evolution of the hydrostatic geopotential. The pressure anomaly is calculated from the hydrostatic and non-hydrostatic densities. Conclusions Figure 1: Schematic view of the refinement and coarsening principles with 2 refinement levels and 3  3 grid cells per block. Figure 2. Distribution of grid points and blocks over the sphere in an orthographic projection centered at (45°N, 0°). The resolution is 2.5°  2.5° non-adapted case, (b) reduced grid case, (c) reduced grid showing adaptations. The library contains modules that: (1)create the block-structured latitude-longitude grid configuration (2)maintain the adjacency information for all blocks at arbitrary refinement levels (3) enable the user to loop over all assigned blocks on a given processor (4) Provide a load-balancing feature that redistributes the blocks among the processors during the adaptive model run The user specifies the algorithms for split and join, and ghost cell operations including interpolation and averaging procedures for the initialization of new blocks and the data exchange algorithms for neighboring blocks at both identical and varying resolutions. Figure 3: Geopotential height field at day 10. The Non-Hydrostatic Dynamical Core One of the most important advances needed in global climate models is the development of models that can reliably treat convection. At the present time, convection is a sub-grid process that must be parameterized. The explicit treatment of convection requires spatial resolutions at which the hydrostatic assumption is no longer valid. Therefore we have coupled a non-hydrostatic code to the hydrostatic model which replaces the hydrostatic treatment if required. The non-hydrostatic code is based on an extension of the hydrostatic code. A mass based Lagrangian vertical coordinate replaces the pressure based Lagrangian vertical coordinate of the hydrostatic code. where is the vertical velocity is the vertically integrated mass per unit area. Our Lagrangian vertical coordinate is defined by, so that the last term in (1) is zero. The mass continuity equation is: (1) The density and vertical velocity is the sum of the hydrostatic part and the anomaly (from the hydrostatic ): Equations for the density and vertical velocity anomaly : Upper boundary condition Along the characteristic equals zero. Here, c s is the speed of sound. Assuming equilibrium and constant density, the above characteristic reduces in one dimension to a condition for the vertical gradient : The use of the above for the upper boundary condition accounts for the coupling between the anomalies for pressure and vertical velocity at the model top. However, the atmosphere is not in equilibrium and the vertical density gradient is far from being zero. Starting from an initial non-hydrostatic density (or pressure) perturbation, we simulated the hydrostatic adjustment process applying the above equation at the model top. The model failed to reach hydrostatic equilibrium, and strong reflections occurred at the model top. The model was numerically unstable. To take into account the transitional character of the atmosphere, we formulated an upper boundary condition based on time derivatives : With this condition the model is much better behaved. Since this equation assumes constant density and doesn't do a full backward integration along the characteristic, small reflections of sound waves still occur at the model top (see Figure 4a). To ensure that hydrostatic equilibrium is always reached, we added a damping term for for the pressure anomaly (Figure 4b). We also implemented a semi-implicit scheme which remains stable at longer time steps and damps the fast-traveling sound waves (Figure 4c). Figure 4: Propagation of a density anomaly in the non-hydrostatic code. The anomaly is initiated near 5 km and propagates vertically where it is reflected at the upper boundary. (a) shows an explicit time step procedure with no damping. (b) shows the explicit model with damping added. (c) shows an implicit version with no damping.